Lemma: Exact stationary and quench functions

For physical waiting time $w\ge 0$ and lag $\ell \ge 0$,

Consequently,

Proof. The first two equations in (eq:two-actions) form a closed two-dimensional linear system. Thus

with the indices interchanged for $P_{\ell }{f}_2$.

Under the uniform start, $m(w)=\mathbb{E}[z(\sigma (w))]$ solves

Hence $m(w)=a(1-e^{-2w})$. Under the invariant law, ${\pi }_a[z]=a$.

Condition on the configuration at waiting time $w$, multiply the displayed semigroup formula by the corresponding initial spin, and average. Both coordinates give the same answer. This proves (eq:two-stat) and (eq:two-quench). Insert $w=Ts$, $\ell =Tt$, multiply by $v$, and subtract.